You are here

Question 2. Proof, Validation, and Trains of Thought

RESEARCH QUESTION

2. Proof, Validation, and Trains of Thought

September 19, 2000

by Annie and John Selden

When a mathematician reads a proof
of a theorem in order to be certain it
contains no errors and actually
proves the theorem, he/she
typically does much more than
read in the ordinary sense.
Questions are asked and answered.
Some of these are explicit in that they
are clearly articulated in inner speech.
Others might be outside her/his main
focus of attention and
consist of recognizing a
familiar situation and judging
whether it has been properly
handled. For example, when something
is to be proved about every number
x and a portion of the proof
begins "Let x be a number . . . ,"
x should neither have occurred
before nor depend on anything that
has occurred before. Most
mathematicians would notice this,
but not focus on it or explicitly ask themselves
(in inner speech), "Is x
arbitrary?" However, any hint that
x might not be arbitrary, would lead to a
careful examination of that portion of the proof.
In addition to asking and answering
questions, a mathematician might recall
definitions and theorems used in the proof,
and their application to the proof
might be examined. Occasionally, they
also use visualization or construct entire
subproofs.

This process may be entirely mental, and
hence not directly observable by someone else.
Furthermore it seems to require considerable
focus of attention and much
short-term memory (Baddeley, 1995).
Thus even a mathematician's self-observations
regarding details may be unreliable and fade quickly
from memory.

We think most mathematicians are
familiar with this process and clearly
distinguish it from ordinary
reading, but they may not have a name for
it. We call it validation.
When one attempts to articulate the validation
process explicitly, the result is often
much longer than the original proof.
We have done this for a short calculus proof
(see Appendix I, Selden and
Selden, 1995).

Because validating a proof for oneself
appears to be an essential part of constructing
it, we view validation as part of the
implicit curriculum. Furthermore, the
reasoning involved appears to be
similar to that required for checking
the correctness of problem solutions.
Moreover, problem solving is emphasized in
the NCTM Standards for school
mathematics. Thus, acquiring the
ability to reliably validate proofs should be
an important part of the education of
preservice teachers, especially secondary
teachers.

There are a number of interesting,
uninvestigated questions regarding validation.
Here we focus on just one. Simply
put: When does a mathematician stop reading?
When is he/she satisfied that the proof is
correct? The validation process
may end when the validator becomes conscious
of a "feeling of understanding and correctness."
However, some mathematicians do not experience this,
and do not conclude the validation process,
until they have reviewed the
entire proof in a single train of
thought. They will go over
the proof from the beginning several
times until they are satisfied. We
mean by a single train of
thought that the validator focuses on
the proof, or a supplemented version
of it, in an uninterrupted way from
beginning to end. Activities such
as constructing subproofs or recalling
complex definitions could break the
train of thought and lead to starting over.
However, a familiar, more or less automated
action, such as drinking a cup of
coffee, probably would not.

Our conjectured
explanation for this phenomenon is
that the validator is trying to be
sure there are no errors in various,
possibly overlapping, "chunks" of the
proof. This often cannot be done in a
simple linear reductive way, e.g.,
dismissing the first half of the
proof as finished and requiring no
further attention, because although
some possible errors are restricted to
a small part of the proof, others are
more global in nature. We
expect this process would require almost
all of one's short-term memory. A
break in one's train of thought could
put an additional, and unnecessary,
burden on short-term memory. This
would increase the chance of errors
of omission, i. e., neglecting to
check something. We are not
suggesting that mathematicians
consciously justify their personal
style of validating proofs according
to this explanation. Rather we suggest that
through experience they may come to
conclude their validations by reviewing
proofs in an unbroken train of thought
because this provides increased reliability.

Question I. Do many
mathematicians commonly end
validations as we have described?

A partial answer might be obtained
by interviewing one, or more,
collections of mathematicians.
By describing them and the results of
the interviews, "intellectual group
portraits" could be produced. For
example, one might interview the
mathematicians of a small department,
or those that are producing research
papers, or those that teach courses
requiring them to evaluate student
proofs. In order to get independent
information from such mathematicians
it might be good if they did not
discuss validation before the
interviews. Also, as subjects
occasionally shift their comments
towards what they think an
interviewer expects to hear, it
might be helpful to avoid describing
the exact purpose of an interview
until its conclusion. Given the
opportunity, mathematicians might
even spontaneously describe the way
they conclude validations and this
could be followed up in subsequent
interviews. Finally, because fine
details of one's own thinking are
hard to notice or remember,
it might be helpful to ask each
mathematician to validate a proof shortly
before the interview.

Question II. Assuming
mathematicians conclude their
validations with an unbroken train
of thought, do
students at various levels do
something similar?

This question might also be partially
answered using interviews, perhaps
starting with middle-level
undergraduates at the end of a
transition course. A "think aloud"
validation might also be useful, but
the length or complexity of the proof
might influence whether multiple passes
through the proof were deemed
necessary.

Question III. Does concluding a
validation by reviewing the entire
proof in an unbroken train
of thought provide more reliability? Does a
mathematician or student who
concludes a validation in this way find
more errors?

One approach to partially answering
this question would be to collect a
set of purported "proofs" containing
a variety of errors, and then to search for
students who can find differing numbers
and kinds of these errors. It might turn
out that students who conclude their
validations with an unbroken train of
thought find more errors, or more
errors of a certain kind. The
students should be fairly accomplished
and the "proofs" moderately long or complex
because there are many situations,
unrelated to this, in which
inexperienced students fail to find
errors. Another approach would be to have
students do think aloud validations and look
for errors they discover when reviewing
the proof from beginning to end. Such errors
might not have been found had such a
review not been undertaken.

The explanation conjectured above is reminiscent
of some recent observations made by those studying
speech-driven computer interfaces.
Word-processor users (i.e., typists) can
often work considerably faster when giving
voice commands such as "italic," than when
using a mouse. However, when asked to
reproduce some mathematical symbols, one
group of typists was slower and had to look
back more often when asked to view the symbols,
say "page down" to the computer, and then type the
symbols. Speech, it turns out, is a heavy user
of short-term memory. Even speaking two words
interfered with remembering the mathematical
symbols (Schneiderman, 2000). Since much of
validation involves inner speech, one
might similarly expect an (inner speech)
interruption in one's train of thought to
interfere with short-term memory and make
a review of earlier work necessary.

The ideas of trains of thought and other
aspects of consciousness have been
investigated (James, 1910), but
largely neglected for some time.
Psychologists, in particular, seem to
have regarded such ideas as difficult to
investigate, but they appear to be
amenable to the kinds of qualitative techniques
used in mathematics education research. For
an example of a more recent paper,
including a discussion of "fringe,"
i.e., the phenomena in consciousness that
are not focused upon, see Mangan (1993).